To pattittrm, tnto oil ttme 2 Srciner
1. lntroductian
1 refer to 131 for the history of the problem. which goes back to 1850; in fxt, Caylcy [ I] showed that there is no packing of arder 7. The existence of a packi!ng of order 9 was discovered by Kirkman 141, and rectiscsvcred ?~erdI t imcs; but, until very recentJy , no packings of other r:>rders had been constructed. Howz;?ver, Schreiber [ 5 j and Wilson [ 7 ] have in&qxndenUy discovered a construction, for a packing of order p+ 2, which works whcncver the prime factors of y are all congruent to t 7 module 8, J had dealt with the eases where 13 = 23, 31, 47 before hearing crsf thi!t construction. Again, Teirlintik [ 61 has shown how to constrtrc t a packing of order 3u, when a packing of order u is given : J have not, accoE:dingIy, studied any orders 3u to which his result could be ap-plied+ Let us identify p of the crlement~ of Y with the integers mrrdulo p, and dcnoae rhe o&r fwcr elemr*nls i,)r A and 8. Let X be the permutstian of Y which Icaves A and B fixe& but adds I mod p to each integer. Therr A ggncrates a group, L ~iy, ofp permutaticlns of 8': and L stfso acts a5 a group of permutal ions an the set T af triads. Since p is divisible neither by 3 nor by J, lhe orbits into whiI:h L partitions T must all ;be of length p. ?j;uppose, then, that we havqz managed to co;nciCruct D Steiner triple qs;:itm S, takbrg just one triad from each orbit of 1:; It.8 S, be a Stiriner tri#e system. namely the image of 5 under X* , where s = 0. I+ . . . l p ... 1.
Then we sex5 al once that {2J,; will 'be a packing.
' But, when a solution to the pro&m has been found, o:?e dif@ufty remains: how iis the !oiution to be presented. so that a r&der may check that the conditions have been satisfied? It is thesome to pick ail the unordered pain out of a iist of t,r?&s. so as to verify that each appears just once. and that the list is 3 Steimr triple system. I therefore waste ' wd for a paif Iw, w + d) of irltegers mod p1 wheqti it is natwaS to take d froc2 the set D givers by 0 < d < #p. A triad of jintegers can be written